Mathematical study of the small oscillations of a finite cylindrical column liquid-gas under zero gravity

Hilal Essaouini,Pierre Capodanno

doi:10.14419/gjma.v7i1.26540

Abstract

This paper deals with the mathematical study of the small motions of a system formed by a cylindrical liquid column bounded by two parallel circular rings and an internal cylindrical column constituted by a barotropic gas under zero gravity. From the equations of motion, the authors deduce a variational equation. Then, the study of the small oscillations depends on the coerciveness of a hermitian form that appears in this equation. It is proved that this last problem is reduced to an auxiliary eigenvalues problem. The discussion shows that, under a simple geometric condition, the problem is a classical vibration problem.Â Â

Highlights

The problem of the small oscillations of an incompressible inviscid liquid under zero gravity, in which the surface tensions determine the character of the motion, is very important in the experiments in space laboratories
The authors study the small oscillations of a system formed by a cylindrical liquid column bounded by two parallel circular rings, the liquid being anchored at the external rim of the rings, and an internal cylindrical column constituted by a barotropic gas, under zero gravity, schematizing an air bubble inside the liquid
The study of the spectrum of the problem depends on a hermitian sesquilinear form that appears in the variational equation and that represents the virtual work of the surface tension forces

Summary

Introduction

The problem of the small oscillations of an incompressible inviscid liquid under zero gravity, in which the surface tensions determine the character of the motion, is very important in the experiments in space laboratories. It is well-known that, in fabrication process under microgravity conditions, such as crystal growth, the oscillations of the free liquid surface are often detrimental effect on the product. This problem has been widely studied since many years by numerous researchers (Moiseyev 1968; Bauer 1982-1989-1993; Morand 1992; Kopachevskii 2001; Langbein 2002; Capodanno 2001-2003). They show that, if we discard translations of the gas column orthogonal to its axis, and under a very simple geometric condition, the problem is classical vibration problem

Position of the problem

Equations of the liquid

Variational formulation of the problem

This form is obviously symmetrical and continuous